Part II, Free actions of compact groups on C*-algebras

Abstract

We study a simple class of free actions of non-Abelian groups on unital C * -algebras, namely cleft actions. These are characterized by the fact that the associated noncommutative vector bundles are trivial. In particular, we provide a complete classification theory for these actions and describe its relations to classical principal bundles.

Part III, Free Actions of Compact Quantum Groups on C^*-Algebras

“…In Section 4 we show that every free compact C * -dynamical system gives rise to a so-called factor system and that free compact C * -dynamical systems can be classified up to equivalence by their associated factor system (Theorem 4.4). This extends the results presented in part 2 of this series [28], which deals with the particular class of cleft actions. Moreover, we give a characterization of cleft actions in terms of their factor systems.…”

“…However, as the following lemma shows, those choices only effect the factor system by a conjugacy with partial isometries: Lemma 4.3 (cf. Lemma 5.5 and Theorem 5.6 in [28]). For a factor system (H, γ, ω) of a free compact C * -dynamical system (A, G, α) with fixed point algebra B the following assertions hold:…”

“…The present paper is a sequel of [27] and [28], where we studied free actions of compact Abelian groups and so-called cleft actions, respectively. To be more precise, we achieved in [27] a complete classification of free, but not necessary ergodic actions of compact Abelian groups on unital C * -algebras.…”

“…For free actions of non-Abelian compact groups the bimodule structure of the corresponding isotypic components is more subtle. For this reason we concentrated in [28] on a simple class of free actions of non-Abelian compact groups, namely cleft actions. Regarded as noncommutative principal bundles, these actions are characterized by the fact that all associated noncommutative vector bundles are trivial.…”

Noncommutative coverings of quantum tori

“…In the present article our investigation revolves around the concrete class of quantum tori, which have been studied by many authors (see e. g. [6,7,8,20]) and have motivated some important work in algebraic K-theory such as the Pimsner-Voiculescu exact sequence for crossed products. By using ideas and methods from [22,23], we are able to construct examples of noncommutative coverings of quantum tori and to characterize all noncommutative coverings of quantum tori that are connected in a reasonable sense. More detailedly, the paper is organized as follows.…”